I have this general formula for the propagator.
D(x)=−i∭d3k(2π)32ωk[e−i[ωkt−→k⋅→x]Θ(t)+ei[ωkt−→k⋅→x]Θ(−t)]
I am supposed to verify that it decays exponentially for spacelike separation. To impose this condition I use t=0 and x>0. The symbols →k and →x are 3-vectors and ωk=+√→k2+m2. Also, for the Heaviside step function Θ(0)=0.5.
This is problem 1.3.1 in Zee's QFT in a Nutshell book, and the solution to this problem is in the back of the book. He writes that the first step should look like
D(x)=−i∭d3k(2π)32√→k2+m2e−i→k⋅→x
but when I plug it in the values for x and Θ, I get
D(x)=−i∭d3k(2π)32√→k2+m212[ei→k⋅→x+e−i→k⋅→x]=−i∭d3k(2π)32√→k2+m2cos(→k⋅→x)
What is the reasoning behind throwing out one of the exponentials instead of using both of them? I haven't done my integral yet, so I don't know if it still produces exponential decay, but it seems like won't, so I want to know what the discrepancy is before I start.
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